3.15 \(\int \text{csch}(c+d x) (a+b \sinh ^2(c+d x))^2 \, dx\)

Optimal. Leaf size=52 \[ -\frac{a^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b (2 a-b) \cosh (c+d x)}{d}+\frac{b^2 \cosh ^3(c+d x)}{3 d} \]

[Out]

-((a^2*ArcTanh[Cosh[c + d*x]])/d) + ((2*a - b)*b*Cosh[c + d*x])/d + (b^2*Cosh[c + d*x]^3)/(3*d)

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Rubi [A]  time = 0.0657536, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3186, 390, 206} \[ -\frac{a^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{b (2 a-b) \cosh (c+d x)}{d}+\frac{b^2 \cosh ^3(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]*(a + b*Sinh[c + d*x]^2)^2,x]

[Out]

-((a^2*ArcTanh[Cosh[c + d*x]])/d) + ((2*a - b)*b*Cosh[c + d*x])/d + (b^2*Cosh[c + d*x]^3)/(3*d)

Rule 3186

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos
[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rule 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \text{csch}(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (a-b+b x^2\right )^2}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (-(2 a-b) b-b^2 x^2+\frac{a^2}{1-x^2}\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac{(2 a-b) b \cosh (c+d x)}{d}+\frac{b^2 \cosh ^3(c+d x)}{3 d}-\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{a^2 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{(2 a-b) b \cosh (c+d x)}{d}+\frac{b^2 \cosh ^3(c+d x)}{3 d}\\ \end{align*}

Mathematica [A]  time = 0.0339139, size = 104, normalized size = 2. \[ \frac{a^2 \log \left (\sinh \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}-\frac{a^2 \log \left (\cosh \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}+\frac{2 a b \sinh (c) \sinh (d x)}{d}+\frac{2 a b \cosh (c) \cosh (d x)}{d}-\frac{3 b^2 \cosh (c+d x)}{4 d}+\frac{b^2 \cosh (3 (c+d x))}{12 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]*(a + b*Sinh[c + d*x]^2)^2,x]

[Out]

(2*a*b*Cosh[c]*Cosh[d*x])/d - (3*b^2*Cosh[c + d*x])/(4*d) + (b^2*Cosh[3*(c + d*x)])/(12*d) - (a^2*Log[Cosh[c/2
+ (d*x)/2]])/d + (a^2*Log[Sinh[c/2 + (d*x)/2]])/d + (2*a*b*Sinh[c]*Sinh[d*x])/d

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Maple [A]  time = 0.032, size = 50, normalized size = 1. \begin{align*}{\frac{1}{d} \left ( -2\,{a}^{2}{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) +2\,ab\cosh \left ( dx+c \right ) +{b}^{2} \left ( -{\frac{2}{3}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \cosh \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)*(a+b*sinh(d*x+c)^2)^2,x)

[Out]

1/d*(-2*a^2*arctanh(exp(d*x+c))+2*a*b*cosh(d*x+c)+b^2*(-2/3+1/3*sinh(d*x+c)^2)*cosh(d*x+c))

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Maxima [B]  time = 1.04856, size = 138, normalized size = 2.65 \begin{align*} \frac{1}{24} \, b^{2}{\left (\frac{e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac{9 \, e^{\left (d x + c\right )}}{d} - \frac{9 \, e^{\left (-d x - c\right )}}{d} + \frac{e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + a b{\left (\frac{e^{\left (d x + c\right )}}{d} + \frac{e^{\left (-d x - c\right )}}{d}\right )} + \frac{a^{2} \log \left (\tanh \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)*(a+b*sinh(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

1/24*b^2*(e^(3*d*x + 3*c)/d - 9*e^(d*x + c)/d - 9*e^(-d*x - c)/d + e^(-3*d*x - 3*c)/d) + a*b*(e^(d*x + c)/d +
e^(-d*x - c)/d) + a^2*log(tanh(1/2*d*x + 1/2*c))/d

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Fricas [B]  time = 1.85076, size = 1280, normalized size = 24.62 \begin{align*} \frac{b^{2} \cosh \left (d x + c\right )^{6} + 6 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + b^{2} \sinh \left (d x + c\right )^{6} + 3 \,{\left (8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + 3 \,{\left (5 \, b^{2} \cosh \left (d x + c\right )^{2} + 8 \, a b - 3 \, b^{2}\right )} \sinh \left (d x + c\right )^{4} + 4 \,{\left (5 \, b^{2} \cosh \left (d x + c\right )^{3} + 3 \,{\left (8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 3 \,{\left (8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 3 \,{\left (5 \, b^{2} \cosh \left (d x + c\right )^{4} + 6 \,{\left (8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 8 \, a b - 3 \, b^{2}\right )} \sinh \left (d x + c\right )^{2} + b^{2} - 24 \,{\left (a^{2} \cosh \left (d x + c\right )^{3} + 3 \, a^{2} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a^{2} \sinh \left (d x + c\right )^{3}\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) + 24 \,{\left (a^{2} \cosh \left (d x + c\right )^{3} + 3 \, a^{2} \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a^{2} \sinh \left (d x + c\right )^{3}\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) + 6 \,{\left (b^{2} \cosh \left (d x + c\right )^{5} + 2 \,{\left (8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} +{\left (8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{24 \,{\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + d \sinh \left (d x + c\right )^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)*(a+b*sinh(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

1/24*(b^2*cosh(d*x + c)^6 + 6*b^2*cosh(d*x + c)*sinh(d*x + c)^5 + b^2*sinh(d*x + c)^6 + 3*(8*a*b - 3*b^2)*cosh
(d*x + c)^4 + 3*(5*b^2*cosh(d*x + c)^2 + 8*a*b - 3*b^2)*sinh(d*x + c)^4 + 4*(5*b^2*cosh(d*x + c)^3 + 3*(8*a*b
- 3*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(8*a*b - 3*b^2)*cosh(d*x + c)^2 + 3*(5*b^2*cosh(d*x + c)^4 + 6*(8*
a*b - 3*b^2)*cosh(d*x + c)^2 + 8*a*b - 3*b^2)*sinh(d*x + c)^2 + b^2 - 24*(a^2*cosh(d*x + c)^3 + 3*a^2*cosh(d*x
+ c)^2*sinh(d*x + c) + 3*a^2*cosh(d*x + c)*sinh(d*x + c)^2 + a^2*sinh(d*x + c)^3)*log(cosh(d*x + c) + sinh(d*
x + c) + 1) + 24*(a^2*cosh(d*x + c)^3 + 3*a^2*cosh(d*x + c)^2*sinh(d*x + c) + 3*a^2*cosh(d*x + c)*sinh(d*x + c
)^2 + a^2*sinh(d*x + c)^3)*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 6*(b^2*cosh(d*x + c)^5 + 2*(8*a*b - 3*b^2)
*cosh(d*x + c)^3 + (8*a*b - 3*b^2)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c)^2*sinh
(d*x + c) + 3*d*cosh(d*x + c)*sinh(d*x + c)^2 + d*sinh(d*x + c)^3)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)*(a+b*sinh(d*x+c)**2)**2,x)

[Out]

Timed out

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Giac [B]  time = 1.28008, size = 174, normalized size = 3.35 \begin{align*} -\frac{a^{2} \log \left (e^{\left (d x + c\right )} + 1\right )}{d} + \frac{a^{2} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{d} + \frac{{\left (24 \, a b e^{\left (2 \, d x + 2 \, c\right )} - 9 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + b^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{24 \, d} + \frac{b^{2} d^{2} e^{\left (3 \, d x + 3 \, c\right )} + 24 \, a b d^{2} e^{\left (d x + c\right )} - 9 \, b^{2} d^{2} e^{\left (d x + c\right )}}{24 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)*(a+b*sinh(d*x+c)^2)^2,x, algorithm="giac")

[Out]

-a^2*log(e^(d*x + c) + 1)/d + a^2*log(abs(e^(d*x + c) - 1))/d + 1/24*(24*a*b*e^(2*d*x + 2*c) - 9*b^2*e^(2*d*x
+ 2*c) + b^2)*e^(-3*d*x - 3*c)/d + 1/24*(b^2*d^2*e^(3*d*x + 3*c) + 24*a*b*d^2*e^(d*x + c) - 9*b^2*d^2*e^(d*x +
c))/d^3